Copyright © Cengage Learning. All rights reserved. 3 Exponential and Logarithmic Functions

Copyright © Cengage Learning. All rights reserved. 3.3 Properties of Logarithms

3 What You Should Learn Rewrite logarithms with different bases. Use properties of logarithms to evaluate or rewrite logarithmic expressions. Use properties of logarithms to expand or condense logarithmic expressions. Use logarithmic functions to model and solve real-life problems.

4 Change of Base

5 Most calculators have only two types of log keys, one for common logarithms (base 10) and one for natural logarithms (base e). Although common logs and natural logs are the most frequently used, you may occasionally need to evaluate logarithms to other bases. To do this, you can use the following change-of-base formula.

6 Change of Base One way to look at the change-of-base formula is that logarithms to base a are simply constant multiples of logarithms to base b. The constant multiplier is

7 Example 1 – Changing Bases Using Common Logarithms a.Log 4 25  2.23 b. Log 2 12  3.58 Use a Calculator. Simplify.

8 Properties of Logarithms

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10 Example 3 – Using Properties of Logarithms Write each logarithm in terms of ln 2 and ln 3. a. ln 6 b. ln Solution: a. ln 6 = ln(2  3) = ln 2 + ln 3 b. ln = ln 2 – ln 27 = ln 2 – ln 3 3 = ln 2 – 3 ln 3 Rewrite 6 as 2  3. Product Property Quotient Property Rewrite 27 as 3 3 Power Property

11 Rewriting Logarithmic Expressions

12 Rewriting Logarithmic Expressions The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of algebra. This is true because they convert complicated products, quotients, and exponential forms into simpler sums, differences, and products, respectively.

13 Example 5 – Expanding Logarithmic Expressions Use the properties of logarithms to expand each expression. a. log 4 5x 3 y b. ln Solution: a. log 4 5x 3 y = log log 4 x 3 + log 4 y = log log 4 x + log 4 y Product Property Power Property

14 Example 5 – Solution Rewrite radical using rational exponent. Power Property Quotient Property cont’d

15 Rewriting Logarithmic Expressions In Example 5, the properties of logarithms were used to expand logarithmic expressions. In Example 6, this procedure is reversed and the properties of logarithms are used to condense logarithmic expressions.

16 Example 6 – Condensing Logarithmic Expressions Use the properties of logarithms to condense each expression. a. log 10 x + 3 log 10 (x + 1) b. 2ln(x + 2) – lnx c. [log 2 x + log 2 (x – 4)]

17 Example 6 – Solution a. log 10 x + 3 log 10 (x + 1) = log 10 x 1/2 + log 10 (x + 1) 3 b. 2 ln(x + 2) – ln x = ln(x + 2) 2 – ln x Power Property Product Property Quotient Property Power Property

18 Example 6 – Solution c. [log 2 x + log 2 (x – 4)] = {log 2 [x(x – 4)]} = log 2 [x(x – 4)] 1/3 cont’d Power Property Product Property Rewrite with a radical.

19 Application

20 Example 7 – Finding a Mathematical Model The table shows the mean distance x from the sun and the period y (the time it takes a planet to orbit the sun) for each of the six planets that are closest to the sun. In the table, the mean distance is given in astronomical units (where the Earth’s mean distance is defined as 1.0), and the period is given in years.

21 Example 7 – Finding a Mathematical Model The points in the table are plotted in Figure Find an equation that relates y and x. Figure 3.22 cont’d

22 Example 7 – Solution From Figure 3.22, it is not clear how to find an equation that relates y and x. To solve this problem, take the natural log of each of the x- and y-values in the table. This produces the following results.

23 Example 7 – Solution Now, by plotting the points in the table, you can see that all six of the points appear to lie in a line, as shown in Figure To find an equation of the line through these points, you can use algebraic method. Choose any two points to determine the slope of the line. Using the two points (0.421, 0.632) and (0, 0) you can determine that the slope of the line is cont’d Figure 3.23

24 Example 7 – Solution By the point-slope form, the equation of the line is Where Y = ln y and X = ln x.You can therefore conclude that cont’d